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Speed of random walk on dynamical percolation in nonamenable transitive graphs
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abstract
Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $\mu>0$, and following the refresh, each edge is open independently with probability $p$. The random walk traverses $G$ only along open edges, moving at rate $1$. In the critical regime $p=p_c$, we prove that the speed of the random walk is at most $O(\sqrt{\mu \log(1/\mu)})$, provided that $\mu \le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $\mu)$, while in the subcritical regime $p<p_c$, the speed is of order $\mu\wedge 1$.
Forward citations
Cited by 2 Pith papers
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Mixing times of spin systems on dynamical percolation
For p below the critical percolation probability and sufficiently small λ, the mixing time of nearest-neighbor Glauber dynamics on dynamical percolation is Θ(log N / λ) on the d-dimensional torus.
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Biased random walk on the critical curve of dynamical percolation
For dynamical percolation in Z^d, the paper derives the e^{-2λ} term in the speed expansion and proves that on the critical curve μ²=p(1-p), d≥2, the speed is eventually increasing in the bias.
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